Mathematical Definitions
First things first, I'm familiar with only high school math so do cut me some slack when you think it's appropriate.
A short prelude: I'm familiar with the term totaled with respect to vehicles. In a layperson's world if a car is totaled, there's no point in repairing it; it's gonna hurt, real bad, to pay for that. That was about all I knew of the concept !
Agent Smith visits Wikpedia: I come across a mathematical definition of totaled: The cost of repairs + The cost of the vehicle as scrap > The initial cost of the vehilce. A precise and very sensible definition by all accounts. This, to me, is the power of math(ematical definitions). It makes complete sense, is clear, and calculable, making life, as it were, easier for everyone.
Given the above, from what I found out, it's as plain as the noses on our faces that definitions should be mathematized.
A penny for you thoughts...
A short prelude: I'm familiar with the term totaled with respect to vehicles. In a layperson's world if a car is totaled, there's no point in repairing it; it's gonna hurt, real bad, to pay for that. That was about all I knew of the concept !
Agent Smith visits Wikpedia: I come across a mathematical definition of totaled: The cost of repairs + The cost of the vehicle as scrap > The initial cost of the vehilce. A precise and very sensible definition by all accounts. This, to me, is the power of math(ematical definitions). It makes complete sense, is clear, and calculable, making life, as it were, easier for everyone.
Given the above, from what I found out, it's as plain as the noses on our faces that definitions should be mathematized.
A penny for you thoughts...
Comments (26)
At your level you are basically learning about tools and preparation. It has nothing to do with mathematics. Most people have no idea what mathematics is and I myself didn’t realise until I was way into my thirties.
In fact, I would write the opposite! If you know high school math you know it all, basically.
You mean total loss?
Math is just a language. It is said to be the language of nature but nature only speaks it if you force it. It's no unìversal language.
Ok, so mathematise the definition for
Like I said though, it is not exactly easy to arrive at a vague understanding and I didn’t realise that merely manipulating numbers (such as what is done is high school and first year of university) is equivalent to calling yourself an artist because you have bought a set of paints.
Having the the ’equipment’ necessary to carry out a task does not mean the task is automatically carried out. That is the best analogy I can think of right now … other than to say it truly is pure magic! :)
How do you know that? I did my share of scattering amplitude calculations or cross sections. Their derivation from path integrals is basically summing.
Quoting I like sushi
This sounds like an idea that is as close to zero as it can get without actually being zero. Is that a thing in maths?
Ha! Yes. The infinitesimal idea. Infinitely close but still something. There will always stay a distance to zero. How can that be? If we approach zero, the point zero and we zoom in, there will always be a space between where we are and zero. At which distance the differential is an infinitesimal? When they touch? But then it's a point. Is an infinitesimal a point? No. Its an infinitely small difference. A tiny 2d linepiece. But if its a linepiece and there a distance? Yes. Infinitely small. Some concept!
Quoting Banno
Let me take a stab at it:
1. Woman: A certain minimum level of the hormones estrogen/progesterone in your brain's chemical milieu.
2. Pessimism: Something similar to the above but involving very different (neuro)chemicals.
3. Depression: This, I'm certain, you already know to be a deficiency of the neurochemical serotonin.
4. Atheism: Possibly, there's a skepticism molecule and it's interacting with a logic molecule. :grin:
5. Cause: Correlation coefficients of unity/1. Of course you knew that!
What sayest thou, o wise one?
:sad:
Quoting I like sushi
What is mathematics?
A big issue as depression it is not as simple as a "deficiency" on neurochemical serotonin. This is probably just the scientific explanation but goes this illness goes further. I cannot count or "mathematise" why a depressed citizen ends up killing himself...
We must try before we die!
Yes, Suicide is the last thing I do. But depression being caused by serotonin deficiency overlooks, in my optimist opinion, the cause for the deficiency, which in a broader context can be seen as an effect of depressing causes. Somehow, the view that neurotransmitter shortage is the cause has it backwards. Depressing circumstances can do the trick.
-Woman
A Sum of infinite complex vectors, with time-variable lengths and rotating with time dependent angular velocities, similar to, but not to be confused with Fourier series:
[math]c_1(t)e^{i\omega _1 t}+ c_2(t)e^{i\omega _2 t} +c_3(t)e^{i\omega _3 t} +...c_n(t)e^{i\omega _n t}... c_{\infty}(t)e^{i\omega _{\infty}t} [/math]
Depression:
The minimum of the God Function, of which sights have been reported. Its exact shape remains largely unknown. Progress is being made though, and probably within the current decade the problem is solved. Breakthroughs have recently been achieved.
Religion:
The God Function
Atheism:
The Hermitian conjugate of the God Function. Though it has been conjectured It's the inverse. Depends on the God Function. Extremist atheists consider the minus symbol sufficient.
Pessimism:
The constant function [math]f(x)=-{\infty}[/math]
Cause:
This one is kind of tricky. There are lots of thoughts about cause and effect. Let's take the physical approach. A force causing an irreversible increase in entropy. Which means every process. Even the seemingly reversible process of two particles interacting by a coupling to a mediating virtual field. Force is the causing factor while initial conditions determine if time goes forwards or backwards. If cause precedes effect or effect cause. Cause cannot exist without effect. So let's define cause as the generator of the effect, separated in time. The most general cause is a non-local one. The particular local cause being a limit of the non-local cause. How can we define the generator of cause? How does matter forces, causes, other matter to change velocity? By coupling to the intermediary virtual condensate, omnipresent in space. How does matter couple to it? By means of charge. Electric charge couples to the virtual photon field. The virtual fields couple, again because of charge, couple to other virtual fields, giving rise to higher order interaction. For example, the virtual photon field couples to the virtual electron field only, which again can couple to the virtual photon field, which again can couple to the virtual electron field, etc. The changes in momentum of two charged particles is caused by all this virtual mediating. The Feynman diagrams represent all this virtual intermediary happenings. A first order diagram by coupling to a virtual intermediary photon (in popular prose referred to as a photon exchanged, which is a rather inaccurate description, as is the short appearance of particle/antiparticle pairs which is actually just one particle rotating). Second order diagrams by permitting the virtual photon to interact with a virtual electron (closed loop) or including an extra virtual photon. Third order by permitting an extra virtual electron again or permitting the previously added virtual electron to couple to a virtual photon. Etcetera, etcetera. This procedure is supposed to go on and on, giving rise to divergent integrals, which are solved by renormalizing. The supposed(!) point-like nature of fundamental matter is the root of the problem. Circumvent this problem (as is only partially done in string theory), and no renormalization needs to be done (Dirac called it a stop gap procedure and Feynman thought it a magic joke).
So, how can we describe a cause with math? Good question! I'll leave the answer to fellow philosophers. What has to be included? Force or interaction. Time running in one direction. Which could be backwards but then the effect precedes the cause, i.e., the effect becomes the cause, while the cause becomes the effect. There should be charges included. And mass. Velocity. Is gravity actually a causing force? Good question. Shall we make an effort?
Does that mean you think about it? This emoticon always confuses me. I rub my chin once in a while too, but why? I nibble a pen(cil) too. Kojak sucked a lolly. You can stick a straw in your mouth. Why men do this? To show they can eat you raw? Should this emoticon be included? Where can I introduce a new one? The pencil nibbler. Somehow similar to this one.
:up:
In fact, as fundamental tools it has everything to do with mathematics. The algebra and trigonometry and perhaps basic calculus you learn is useful throughout the incredibly diverse realm of mathematics.
Still, the woven contours are fascinating! It's all there in the first lines. It's basically rotating vectors in the complex plane and filling in each others previous (n-1) values in their n state, bringing a P[0.1] along, differentiate (taking the ratio of differentials, which are somewhat strange math entities, but comprehensible) the e powers wrt t, and filling in numbers. But to contemplate such intricate is something else. I can't remember I have seen these when I did complex analysis. Did residues and contour integration, etc. but this seems new. Or is it a normal thing in the field. Did you turn it into a visual? :chin:
Coupled Differential Equations
There is a tremendous amount of complex variable theory that is not covered in a typical course (which I taught a number of times). I got into my specialty over a half century ago via complex valued continued fractions, which led to a general study of infinite compositions of complex functions.
Yes, I extracted the visuals.
:up:
Tried an analytical solution once of a particle falling in a non-uniform gravitational field, with varying g. It got larger and larger... The 32x32 supermagic Franklin square was fun to do! Started from scratch, but it gave me good understanding. :nerd:
Did you invent those contours eating each other? The zn and ksi n? And the eta connecting them?
Just examples of little interest apart from the imagery.
[quote=Neil deGrasse Tyson]So, we're just sacks of chemistry? :chin:[/quote]
Quoting Haglund
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Quoting Haglund
What's a/the God Function. Cool name nevertheless...is it like the God Particle, a step towards a The Truth?
Quoting Haglund
Nec caput nec pedes.
Quoting Haglund
Good one!
Quoting Haglund
I see causality as a pattern (duh!) and that puts it within the mathematical domain. Legend has it that mathematicians can always explain patterns.